MATH/208 Agenda WorkShop 4

1. Questions about Workshop 4 material                           10 Min
 
10 Minute Break
 
2. Submitting of Learning Team Assignments (And Logs)         5 Min
 
3. Learning Team In-Class Discussion Questions        5 Min

I. Estimations and Modeling

   o Complete the following in your team meeting and be prepared to report your results and discussions to the class:

   o Complete parts 1-4 from the Group Activity at the end of section 3.3 in 
   	Ch. 3 of the text.

o   Discuss any concept that a team member is having a difficult time 
	understanding.

o   Examine the importance and applicability of this week's concepts to each 
	team member and to society in general.

o   Plan and work on team assignments or projects.

    o Give a brief update to the class in which you present your team's 
    	solutions from this exercise. Be sure to describe how this exercise 
	applied to the weekly concepts.

    o Prepare to update your instructor on your team's progress on Learning 
    	Team assignments.

II. Scatterplot and Scope

    o Complete the following in your team meeting and be prepared to report
		your results and discussions to the class:

    o Complete parts 1-6 from the Group Activity at the end of section 8.1 in 
	Ch. 8 of the text.

o   Discuss any concept that a team member is having a difficult time understanding.

o   Examine the importance and applicability of this week's concepts to each 
	team member and to society in general.

o   Plan and work on team assignments or projects.

    o Give a brief update to the class in which you present your team's 
    	solutions from this exercise. Be sure to describe how this exercise 
	applied to the weekly concepts.

o   Prepare to update your instructor on your team's progress on Learning Team 
	assignments.

III. Slope

    o Complete the following in your team meeting and be prepared to report 
    	your results and discussions to the class:

    o Complete parts 1-5 from the Group Activity at the end of section 3.5 in 
    	Ch. 3 of the text.

o   Discuss any concept that a team member is having a difficult time 
	understanding.

o   Examine the importance and applicability of this week's concepts to each 
	team member and to society in general.

o   Plan and work on team assignments or projects.

    o Give a brief update to the class in which you present your team's 
    	solutions from this exercise. Be sure to describe how this exercise 
	applied to the weekly concepts.
 
	Other Discussion Questions
	Other Discussion Questions

	 o Using the readings in Ch. 3 of the text, identify and explain at 
	 least one real-world application of algebraic concepts for one of the 
	 following areas: business, health and wellness, science, sports, and 
	 environmental sustainability. Do you think it is easier to relate this
	 concept to one of these areas over any other? Explain why.

	 o Imagine that a line on a Cartesian graph is approximately the 
	 distance y in feet that a person walks in x hours. What does the 
	 slope of this line represent? How is this graph useful? Provide 
	 another example for your colleagues to explain.

         o Can one line have two slopes? Explain how or why not.

	 o What is the difference between a scatterplot and a line graph? 
	 Provide an example of each. Does one seem better than the other? 
	 In what ways is it better?

	 o If a line has no y-intercept, what can you say about the line? What 
	 if a line has no x-intercept? Think of a real-life situation where a 
	 graph would have no x- or y-intercept. Will that always be true for 
	 that situation?

	 o Explain the concept of modeling. How does a model describe known 
	 data and predict future data? How do models break down? Can you think 
	 of an example?

	 o What are the differences among expressions, equations, and functions? 	Provide examples of each.

       
 
4. Quiz 4                                                        60 Min
 
5. Topics and Objectives Week 4                                    5 Min
 



 

6. Terms, Factors, Coefficients, Exponents, and Mathematical Operations of Polynomials          40 Min

        o Addition, subtraction, multiplication, and division of whole numbers 
          and fractions:
        Addition:  a + b = b + a        (a + b) + c = a + (b + c)
        3     2
       --- + --- 
        4     3

        Subtraction:  a - b different from b - a: 
	                                                no commutative property
                                                  no associative property
        3     2
       --- - --- 
        4     3

        Multiplication: ab = ba      (ab)c = a(bc)    c(a + b) = ca + cb

        3     2
       --- * ---  - 2
        4     3


        3     2
       --- + ---(3 + 3*2)
        4     3


        Division: a/b different from b/a:   no commutative property
                                                   no associative property

        3     2    / 1
       --- * ---  / ---
        4     3  /   5


        o Identify a term in an algebraic expression.

        Definition: Term: a number, variable, or both which are 
                                 connected by addition or subtraction 
                                 signs (operators).  Multiple terms in
                                 an expression are separated by + or -
                                 signs 

        ax + by 

        o Identify a coefficient when adding and/or subtracting like terms.

        Definition: Factor: a number, variable, or both which are 
                                 connected by multiplication or division 
                                 signs (fractions).  Multiple factors in
                                 an expression are separated by * or /
                                 signs 

        Definition: Coefficient: In a product, any factor or group of
                                        factors is the coefficient of the
                                        remaining factors. 
        
        ax/by 

        o Identify a factor in an algebraic expression. 

        Definition: Factors of a product:  If 2 or more numbers are
                                                  to be multiplied, each of
                                                  them as well as the product
                                                  of any of them is a factor
                                                  of that product. We usually
						  consider an entire expression
						  as one factor.

        ax + a   [two terms: ax , a  ; two factors a(x + 1)]

        o Laws of Exponents:

	  Negative Exponents:  If a is a real number, not zero, and
	                       n is a positive integer, then:
           -n     1  n              -1    1
	  a   = (---)              a   =  -
	          a                       a


             1        n             a  -n     b  n
          ------- =  a            (---)   = (---)
             -n                     b         a 
            a

          Integer Exponents (a, b not zero; m, n integers):
	  
           0                  m    n    m+n
	  a  = 1             a  * a  = a

           m
          a     m-n            m n    m*n
	  -- = a            ( a )  = a
	   n 
          a
                                         n
              n     n    n      a  n    a
          (ab)   = a  * b     (---)  = ---
                                b        n  
                                        b
       o Multiplicative property of like factors, such as: 

	  x*x = x²

           1    2     1    2      3 + 4     7      1
           _    _     _  + _      -----     -     1-
           2    3     2    3        6       6      6
	  X  * X  =  X        =  X       = X   = X

        o Identify exponents when multiplying and/or dividing like factors.

          2 
        2X  + 3X + 4

          3
        2X  + 4(X-2)


        2X - 1
        ------  + 3X + 4
           4

        o Distributive law and its relevance with the definition of coefficient
                          4    4     4     4
        For example, if 3Y  = Y  +  Y  +  Y , then how is it possible to show 
                 4            4
        that 3(2X  +  1) =  6X   +  3

        without the use of the distributive law?

        o Why is the following identity true?

          4           4
        2X  +  1    2X         1
        -------- =  ----  +   ---
            3        3         3

        o The order of operations in simplifying expressions.
          
          1. Evaluate starting from left to right 
          2. Replace any variables by their specific values, if any.
          3. Eliminate parentheses first by evaluating the expression within
          4. Exponents (powers) and radicals (e.g. square roots)
          5. Multiply and Divide
          6. Add and Subtract

        o Identify rational terms and factors in rational expressions. 
          Addition and Subtraction of rational fractions need a Lowest 
          Common Denominator (LCD) and multiplication and division do 
          not require the lowest common denominator.

                       2                  2
          Compare:   3Z     2Z + 1    4(3Z ) + 5(2Z + 1)
                    ----  + ------  = ------------------
                      5        4               20

          and:         2                 3     2
                     3Z     2Z + 1     6Z  + 3Z
                    ----  * ------ =  -----------
                      5        4          20


        o Identify like terms and common factors.
          
          Definition: Like Terms: Similar terms having the same 
                                         variables as factors with corresponding
                                         variables having the same exponent.
                                 
          Examples:    Good - Like Terms
                         2     2      2
                       4X  + 3X   = 7X
                       3L + 5L = 8L
                       9ab - 2ab = 7ab
                       4(x + y) + 6(x + y) = 10(x + y)

		       No Good - Unlike Terms
                         3     2 
                       4X  + 3X  
                       2x + 3y
                       3ab - 4cd
                       
           
 
8. Learning Teams Exercise                                       15 Min

	     
 
10 Minute Break
 

h.  Discussion Questions:
        o  What is the difference between a term and a factor?
        o  When do you use distributive properties?
        o  What operations are associated with coefficients?
        o  What operations are associated with exponents?

 
9. Exponents and Mathematical Operations with Polynomials          40 Min
 
     o Simplification of a polynomial by using the concept of distributive 
       property and like terms.  For example, simplification of the 
       expressions of the following forms:

     Examples for simplification:

         2   3            2
       3X (2X  - 1) - 2(2X  - 3)

       
           3        4      3
       4(4X  + 1 - X ) + 2X (4 - X)


       1   3         1    3
      ---(X  - 1) + ---(3X  + 2)
       2             2

         4
       3X     2
       --- + ---(3X + 6)
        4     3 


     o Simplification of polynomials using common factors

       Examples:

          2 3   2
       (3X ) * X

          2 3   2
       (3X ) * X
       ----------
            2
          9X

      o Simplification of polynomials with a fractional exponent

        Examples:

            3   1
           --- ---
          2 2   2
       (3X ) * X

          2 3   2
       (3X ) * X
       ----------
            1
           ---
             3
           9X

      d. Multiplication and division of polynomials by using distributive 
         property:
            2          2
         (3X -3X -2)(3X  - 4X -1)


            2          
         (3X -3X -2)/3X

      e. Long Division Of Polynomials:
          2
         X  + 3X + 5                  ____________
	 ----------  same as          | 2
	    X + 1               X + 1 |X  + 3X + 5
         

          3
         X  - 1                   __________________
	 ------  same as          | 3     2
	 X - 1              X - 1 |X  + 0X  + 0X - 1

 
      See Polynomial Division Examples and Procedure
 
      f. Simplifying radicals and rationalizing the denominator:

         sqrt(4) = sqrt(2*2) = 2

	 sqrt(20) = sqrt(4*5) = sqrt(4)*sqrt(5) = 2*sqrt(5)

	    1           1      sqrt(2)      sqrt(2)      sqrt(2)
	 -------  =  ------- * -------  =  ---------  =  -------
         sqrt(2)     sqrt(2)   sqrt(2)     sqrt(2*2)        2

     g. Scientific Notation:

        See: Convert between Real Numbers And Scientific Notation
	or
	Scientific Notation And Worksheet
 
10. Learning Teams Exercise                                      10 Min

 
 

 
10. Workshop Summary of key points                                5 Min

  • 4.1 Use exponents in algebraic expressions
  • 4.2 Apply exponential principles to scientific notation
  • 4.3 Simplify polynomials
  • 4.4 Use the distribution property with polynomials
  • 4.5 Perform polynomial operations
  • 4.6 Use exponents and polynomials in real-world applications.
  11. Preview of Workshop 5 5 Min Review of Final Exam Topics o Order of Operations o Distributive Law o Simplifying and evaluating an expression o Operations with fractions o Solving a linear equation o Solving a system of equations o Solving a linear inequality o Solving a system of inequalities o Formulating word problems into equations o Ratios, fractions, percentages, and their working principles o Graphing of one or two equations or inequalities   12. Multiply By 7 Fast:  

Questions? Robert Katz: arkay@speakeasy.org
Last Update April 23, 2011